Questions tagged [lie-groupoids]
The lie-groupoids tag has no usage guidance.
26
questions with no upvoted or accepted answers
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Are Lie groupoids just groupoids internal to smooth manifolds?
It seems to be common to say "no" - but is this true?
Two weeks ago I asked for a counterexample, but received no replies.
To give some background, let's recall that the difference between ...
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Differentiation of Lie $\infty$-groupoids
I've been trying to understand how to differentiate Lie $\infty$-groupoids to get a Lie $\infty$-algebroid. First of all, I will state the definitions that I'm assuming.
A Lie $\infty$-groupoid is a ...
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Holonomy as a right adjoint, monodromy as a left adjoint
This question about the difference between holonomy and monodromy has an interesting answer by Ronnie Brown.
An excerpt:
So holonomy comes out as a kind of right adjoint, and monodromy as a kind ...
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What are the Newton groupoids from Drinfeld's paper on the Grinberg-Kazhdan theorem?
The paper the Grinberg-Kazhdan formal arc theorem and the Newton groupoids by Drinfeld seems to contain many interesting things which are beyond me. For now, I am trying to get some intuition for the ...
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Example of a groupoid internal to the category of smooth manifolds that is not a Lie groupoid
This questions is about the distinction between:
Lie groupoids: we require source and target maps to be submersions. This implies that the domain of the composition map, $G_1 \;{}_s\!\times_t G_1$, ...
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On Holonomy in (regular) Riemannian Foliations
Right now, I am trying to understanding the role of holonomy fields on Riemannian foliations, which lead me to the following (probably topological) groupoid:
Let $\mathcal{F}\subset M$ be a ...
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KK-witnesses of Gysin maps between differentiable stacks
In 1982 Alain Connes gave the construction of a KK-element $f! \in KK(C(X), C(Y))$ that "witnesses" the fiber integration/Gysin/Umkehr/wrong-way map on topological $K$-theory along a K-orientable map ...
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Averaging over a Weinstein groupoid?
(Not sure if this question belongs here or on m.SE)
For a Lie group, $G$ (of dimension $n$), one can average over the group:
$$
\Gamma = \int_{G} d\mu(g) ~g
$$
(where $d\mu(g)$ is the left-Haar ...
- 423
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Do we have classification (upto Morita equivalence) of Lie groupoids?
Vague question is the following:
Is there a classifcation of Lie groupoids?
Slightly less vague question is the following:
Is there a (short?) list of "types" of Lie groupoids such that ...
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Reference request : Quotient manifold theorem for Lie groupoid action on a manifold
Let $G$ be a Lie group and $M$ be a smooth manifold. Let $G\times M\rightarrow M$ be a smooth map giving a free, proper action of $G$ On $M$. Then, by quotient manifold theorem, we see that there ...
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Is this groupoid a model for the derived fixed-point locus of the free loop space?
In this paper, John Baez and Urs Schreiber define (see Definition 2.16) a Lie groupoid (there called a '2-space') associated to any manifold $M$. In fact it is a bundle of Lie groups over $M$ thought ...
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Degeneration of spectral sequence computing Hochschild cohomology of enveloping algebra of Lie algebroid
Let $L$ be a Lie algebroid on a smooth affine $k$-scheme $X=spec(R)$. Recall that by definition $L$ is a locally free sheaf with the structure of a sheaf of $k$ Lie algebras, so that there exists a ...
user108998
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Couniversality of Lie integration in different categories of manifolds/smooth spaces
A fairly reasonable interpretation of Lie II and Lie III seems to be that the category of Lie algebras is a coreflective subcategory of the category of Lie groups, so that the Lie group integrating a ...
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Lie group (topological group) action on differentiable stack (topological stack)
Let $G$ be a Lie group and $\mathcal{D}$ be a differentiable stack (I am also ok to start with a topological group and topological stack).
I have seen someone mentioning somewhere that the notion of ...
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First thoughts about fundamental group of a topological (Lie) groupoid
I am reading the paper Chern-Weil map for principal bundles over groupoids.
In page number $13$, authors say
let us recall the definition of fundamental group of a topological groupoid.
But, they ...
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answers
141
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Properties of the induced map between inertia stacks
Let $\mathcal X$ and $\mathcal Y$ be (separated) Deligne-Mumford stacks. A morphism of stacks $f:\mathcal X \to \mathcal Y$ induces a morphism between inertia stacks $\tilde f:I\mathcal X \to I\...
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Does the convolution $C^*$-algebra of locally compact Hausdorff groupoids recover back the respective groupoid?
First of all, my knowledge of operator algebras (and functional analysis) is very superficial, so sorry if the answer is actually well-known.
Let $X$ be a locally compact Hausdorff groupoid (or Lie ...
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84
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Examples of strictification of a weak category obtained from a generalisation of a strict category
I have made the following observation (hopefully a correct one) when reading the paper Orbifolds as stacks:
They start with the strict $2$-category category of Lie groupoids, functors, natural ...
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Condition on a Lie groupoid to be represented by manifold/group or an action groupoid
Let $\mathcal{G}$ be a Lie groupoid. I am thinking of following questions.
When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $(G\rightrightarrows *)$ for some ...
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Action of a Lie groupoid on a Lie Algebroid?
Let $\pi:E\longrightarrow M$ be a vector bundle. Then we can associate a Lie groupoid $\mathsf{Gl}(E)\rightrightarrows M$ where $$\mathsf{Gl}(E):=\{E_x\stackrel{lin. isom.}{\longrightarrow} E_y: x, y\...
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Which makes Lie groupoids so nice?
This is a continuation of my previous question.
A) Morphisms in (1') are basically internal anafunctors, their compositions heavily use (and only) pullback/limit.
B) Bibundles in (2) are basically ...
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How to prove the gluing-condition for a pseudogroup induced by an étale Lie groupoid?
Let $G_1\substack{\to \\ \to}G_0$ be an étale Lie groupoid, whose source- and target-maps are denoted by $s$ and $t$, respectively. Let
\begin{equation}
\Psi=\{(t|_U)\circ(s|_U)^{-1}:\text{$U$ is ...
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vote
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Is there an inverse image functor for sheaves on stacks?
I'm interested specifically in an inverse image functor between differentiable stacks, ie. stacks coming from Lie groupoids. Specifically, if I have a morphism of Lie groupoids $H\to G$ and I have a ...
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Idea behind definition of classifying space over an orbifold
Today I was explaining to some one the notion of $\mathcal{G}$ spaces, covering spaces over orbifolds from Orbifolds as Groupoids: an Introduction.
Definition : Let $X$ be a locally compact ...
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adjoint representation of 2-Lie groups
Baez and Crans in their paper on Lie 2-algebras refer to adjoint representations of Lie 2-groups but don't say much, as far as I can tell, except to say that such a representation acts on a 2-Lie ...
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Is there a classifying space for transitive Lie algebroids? If so, what is it?
Let $M$ be a manifold. The data of a Lie groupoid over $M$ is equivalent to the data of a singular foliation $M=\sqcup\mathcal{F}_i$ and, for each $i$, a map (mod homotopy) $f_i:F_i\to BG_i$ (where $...
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